Him shouting at everyone and being unreasonable has nothing to do with your cycle.
You snapping because he's been shouting at you and your daughters has nothing to do with your cycle.
You being annoyed about the unequal distribution of housework for years has nothing to do with your cycle.
You said:
This only started a few months ago when I started getting angry about inequitable distribution of housework.
...so this clearly has nothing to do with your cycle; it's because you asked him to do his fair share of the domestic labour. It's likely that if you stopped asking him for help, and resigned yourself to being the household drudge, the shouting would stop.
The problem here is your husband and his failure to (a) do his fair share of domestic chores, and (b) control his emotions. You are not going insane; he is driving you insane with his unreasonable behaviour.
He feels I don't appreciate that he is trying to take on more.
So you need to appreciate him for "trying"?
But it seems he doesn't appreciate that you have been actually "doing" (all of it) for years. It appears that your husband is "nice" when you keep quiet and accept being the domestic servant; and he becomes less nice when you ask him to do his fair share.
You have a need: "I need others to do their share of the domestic work". But he's not actually doing any - instead he's flapping around making noise and fuss. He may be (subconsciously?) trying to stop you from asking for help by using this behaviour as a deterrent: you ask him to do his share = he kicks off; to avoid him kicking off, you stop asking him to do his share.
Thinking practically, combat the noise, fuss and chaos with structure. Write a list of all the jobs that need doing, and on what days, and then all of you sit down together to create a rota. Allocate jobs to each family member - the girls are old enough to hoover, load/empty the dishwasher, clean the bathroom or make a meal once a week (include tidying their rooms on the rota). Print the rota out, laminate it, and stick it on the fridge so tasks can be crossed off each day/week. Wipe clean at the end of the week and start again. Or use an app.
By giving your husband clear duties, there is no more "appreciate me for trying" - he will either have completed his tasks or not. By giving your daughters clearly defined duties, it will (hopefully) reduce the conflict created by trying to get them to clean their rooms. I'd also tie their chores to their pocket money.
Good luck OP. Remember - you are not going insane. It is understandable to be upset when you are being treated badly.